A line is NOT infinitely divisible. Numbers are. — TheMadFool
At which point we should try to figure out what the ontological facts about time are supposed to have to do with the concept of numbers. — Terrapin Station
Mathematics is the language of the universe — Galileo
Quantification (numbers) is the problem and also the solution.
It's the solution because once we have the numbers we can understand. — TheMadFool
It's just the problem, because there's no reason to believe that time (or space for that matter) works just like our concepts re numbers. — Terrapin Station
One of Zeno's most famous paradox has to do with Achilles never being able to catch a tortoise that's been given a head start in a race because of the impossibility of having to traverse an infinite number of points between the two.
Indeed, mathematics is the science of drawing necessary conclusions about hypothetical states of things, which may or may not match up with any real states of things.The problem is that mathematics is a way that we think about relations. The world isn't required to match that. — Terrapin Station
Again, a continuous line or interval of time does not consist of discrete points or instants at all, but we can mark any multitude of points or instants along it to suit our purposes. In other words, contrary to Cantor, there is a fundamental difference between a continuum and a collection.If we take time to be on a number line how many points of time are there between 1976 and 2019? — TheMadFool
What Cantor got right is that there is likewise a fundamental difference between an infinite collection and a finite collection, such that we cannot reason about them in the same way. The multitude of real numbers between any two arbitrary values is the same, because they can be put into one-to-one correspondence with each other.there are infinite numbers between 0 and 1, but it is intrinsical that there are more numbers between 0 and 2 — Filipe
No, the collection of all combinations of the subjects of a collection--even an infinite collection--is always of greater multitude than that collection itself. The integers and the rational numbers can be put into one-to-one correspondence with each other, but not with the real numbers, because those are of the next greater multitude. There is another multitude greater than that, and another greater than that, and so on endlessly--which is why an infinite collection of any multitude can never be "large" enough to qualify as a continuum.You can put a one to one correspondence between any infinite set, because infinite sets have units. — Gregory
That depends on what you mean by "parts." The portions of a continuum are indefinite, unless and until they are deliberately marked off by limits of lower dimensionality to create actual parts. For a one-dimensional continuum like a line or time, those limits are discrete and indivisible points or instants that serve as immediate connections between portions, but the portions themselves remain continuous--which is why they can always be divided further by inserting additional limits of any multitude, or even exceeding all multitude.Likewise, unless you are speaking of process philosophy, an object must have parts. These can be divided endlessly, so it is neither discrete nor continuous. — Gregory
Because the real numbers correspond to all the possible combinations of rational numbers, and therefore are necessarily of greater multitude than the rational numbers themselves--which are of the same multitude as the natural numbers, along with the even numbers, the odd numbers, the whole numbers, the integers, etc.If I can find all the even numbers but line all the odd numbers with all the whole numbers, why can't I do this with all the real numbers? — Gregory
"Countable" is defined as being of the same multitude as the natural numbers, and thus applies to the rational numbers, the even numbers, the odd numbers, the whole numbers, the integers, etc. "Uncountable" is defined as being of a multitude greater than that of the natural numbers, and thus applies to the real numbers.Nothing has been settled to be countable or uncountable at that point yet — Gregory
Cantor wrongly thought that the real numbers constitute a continuum, but as I noted previously, they can only constitute an infinite collection--one whose multitude is greater than that of the rational numbers. His own theorem proves that there is another collection of even greater multitude, and another greater than that, and so on endlessly. Consequently, a true continuum cannot consist of discrete subjects (like numbers or points) at all.The infinity of the continuum would suggest that all objects have the same infinity, Thus thought Cantor. — Gregory
Right, but that paradox stems from the same mistake of treating discrete points as if they were somehow continuous. It reflects a limitation of such standard models of continuity, which are adequate for most mathematical and practical purposes. Banach-Tarski does not arise in a better model of true continuity, such as synthetic differential geometry (also called smooth infinitesimal analysis).But Banach and Tarski essentially pointed out that this would mean you could take a mountain out of a pea. — Gregory
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